The Estimation of Approximation Error using the Inverse Problem and the Set of Numerical Solutions

TL;DR

This paper introduces a method to estimate point-wise approximation errors in numerical solutions of PDEs using inverse problems and Tikhonov regularization, validated on shock wave interference flows.

Contribution

It proposes a novel inverse problem approach with Tikhonov regularization to estimate approximation errors from multiple numerical solutions on the same grid.

Findings

The method accurately estimates approximation errors in complex flow simulations.

Comparison shows the estimated errors closely match true errors.

The approach is validated on two-dimensional shock wave interference problems.

Abstract

The Inverse Problem for the estimation of a point-wise approximation error occurring at the discretization and solving of the system of partial differential equations is addressed. The set of the differences between the numerical solutions is used as the input data. The analyzed solutions are obtained by the numerical algorithms of the distinct inner structure on the same grid. The approximation error is estimated by the Inverse Problem, which is posed in the variational statement with the zero order Tikhonov regularization. The numerical tests, performed for the two dimensional inviscid compressible flows corresponding to Edney-I and Edney-VI shock wave interference modes, are provided. The analyzed flowfields are computed using ten different numerical algorithms. The comparison of the estimated approximation error and the true error, obtained by subtraction of numerical and analytic solutions, is presented.